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Gaia DR2 has revealed new small-scale and large-scale patterns in thephase-space distribution of stars in the Milky Way. In cylindrical Galacticcoordinates $(R,\phi,z)$, ridge-like structures can be seen in the \vphiR{}plane and asymmetric arch-like structures in the \vphivR{} plane. We show thatthe ridges are also clearly present when the third dimension of the \vphiR{}plane is represented by $\langle z \rangle$, $\langle V_z \rangle$, $\langleV_R \rangle$, $\langle$[Fe/H]$\rangle$ and $\langle[\alpha/{\rm Fe}]\rangle$.The maps suggest that stars along the ridges lie preferentially close to theGalactic midplane ($|z|<0.2$ kpc), and have metallicity and $\alpha$ elementalabundance similar to that of the Sun. We show that phase mixing of disruptingspiral arms can generate both the ridges and the arches. It also generatesdiscrete groupings in orbital energy $-$ the ridges and arches are simplysurfaces of constant energy. We identify 8 distinct ridges in the \gaia{} data:six of them have constant energy while two have constant angular momentum.Given that the signature is strongest for stars close to the plane, thepresence of ridges in $\langle z \rangle$ and $\langle V_z \rangle$ suggests acoupling between planar and vertical directions. We demonstrate, using N-bodysimulations that such coupling can be generated both in isolated discs and indiscs perturbed by an orbiting satellite like the Sagittarius dwarf galaxy.

The GALAH survey and Gaia DR2: Linking ridges, arches and vertical waves in the kinematics of the Milky Way